(note that here we take only the single commutant and \(f\)  has support in \(W_{\mathrm {L}}\) ,
while \({\mathcal {M}}_{S_2}\)  corresponds to \(W_{\mathrm {R}}\) . By the wedge-duality, this is just a matter of convention
(see [1]})).
The triple \(({\mathcal {M}}_{S_2}, U_{S_2}, \Omega _{S_2})\)  is a Borchers triple [2]}.
Furthermore, the modular nuclearity holds if \(S_2\)  is regular and fermionic [3]},
hence the triple is wedge-split in those cases.
